_{2}
, q
_{2}
≥63.72. Then the price is p
_{2}
=105q
_{2}
, and the profit of F
_{2}
has the crest point at q
_{2}
=47.73 as in the previous case that is out of the boundary q
_{2}
≥63.72. When q
_{2}
<63.72, the demand is d
_{2}
=(q
_{1}
+q
_{2}
)/2 and the price is p
_{2}
=73.140.5q
_{2}
. The optimum of the function is at q
_{2}
=60.95 which is inside the region. Based on these profit functions, F
_{2}
selects q
_{2}
=60.95 while discarding the choice before 49.58.
E
. Case ⑤: The F
_{1}
’s best reaction under q
_{2}
=60.95 is also based on the profit functions with respect to q
_{1}
. While the line is not congested, d
_{1}
q
_{1}
≤5, d
_{1}
=(q
_{1}
+q
_{2}
)/2, and q
_{1}
≥50.95. The profit of F
_{1}
under no congestion has a crest at q
_{1}
=49.6, which is not feasible. The attainable optimum point is q
_{1}
=50.95 and π
_{1}
=1725.1 on the boundary case. As checked before, F
_{1}
’s choice under congestion is q
_{1c}
and π
_{1}
=1880.2 larger than 1725.1. Therefore F
_{1}
’s strategic response under q
_{2}
=60.95 is to pick the q
_{1c}
=39.58 discarding the pick before 53.72. This means to return to the case ①, and it will change the situation to cases ②, ③, ④, and ⑤ successively. These choices are not called equilibrium strategy, since they keep changing their choices when opposite player changes. This phenomenon is called “cycling”.
Table 1
shows this cycling by summarizing their strategies.
Summarized best response and cycling of F1and F2’s reactions
Summarized best response and cycling of F_{1} and F_{2}’s reactions
 2.3 Mixed strategy equilibrium
The cycling phenomenon results from the discontinuities in their reaction curves. The transmission line congestion makes the decision space divide into subsets depending on whether they are congested or uncongested. This leads to discontinuities and nondifferentiable profit functions.
The hierarchical optimization of (1)~(6) can be solved easily by the mathematical programming method, once no inequality binds underlying decision space of differentiable functions in the Cournot model. Let the solved quantity parameters be
q
^{*}
. At a pure strategy NE, the strategies of all participants satisfy
where
is the solved quantity parameter of firm
i
,
q_{i}
is the possible quantity parameter firm
i
can choose, and
is the solved quantity parameter set of all participants excluding firm
i
. By unilaterally altering their choices, none of the generating firms can improve their profits.
It may be that none such pure strategies satisfy the definition of Nash equilibrium (7). Instead, the firms may discover that they must play a combination of pure strategies, choosing amongst them randomly. This is a “mixed strategy,” which is specified by the probability distribution of the choice of pure strategies
[7
,
15]
.
Solving the twolevel optimization for the sample case as summarized in
Table 1
gives a mixed strategy Nash equilibrium (MSNE); q
_{2}
=54.89, q
_{1}
=[39.58, 51.83] with probability [0.657, 0.343]. At the NE, the firm, F
_{1}
chooses a mixed strategy consisting of two pure strategies with the probabilities, while F
_{2}
chooses a pure strategy. One of the two choices of F
_{1}
gives rise to congestion on a line, for example q
_{1}
=51.83. The other, q
_{1}
=39.58, on the contrary, makes no congestion. The former can be called a congestion strategy, the latter an uncongestion strategy. F
_{1}
uses more strategic choices, sometimes congestion strategy; other times uncongestion strategy, whereas F
_{2}
takes only one choice.
This paper asserts that the player choosing a mixed strategy plays the role of a leader, and the other plays the role of a follower from the viewpoint of the Stackelberg model. The leading player locates in a receiving area over a congested line, whereas the follower locates in a sending area. The leader, F
_{1}
, is at bus 1on the receiving end in
Fig. 1
. This assumption is very useful in calculating a MSNE, and its usefulness has been verified over a variety of case studies.
3. Stackelberg Model
 3.1 Stackelberg equilibrium at duopoly
It is assumed in Cournot model that all the competitors in a market have equal opportunity. However, in some industries, historical, institutional, or legal factors put the competitors into a differential or inequitable position in the market. For example, the firm that discovers and develops a new product has a natural firstmover advantage.
Heinrich von Stackelberg presented an important oligopoly model in 1934. In the Stackelberg model, one firm acts before the others. The leader firm picks its output level and then the other firms are free to choose their optimal quantities given their knowledge of the leader’s output. The follower’s best response is determined as in the Cournot model that competitors’ output is assumed to be fixed. However, the leader’s action is different from the followers’ actions in Cournot model
[8]
.
The leader picks the output to maximize its profit subject to the constraint that the follower firm chooses its corresponding output using its Cournot best response function
[16
,
17]
. In Stackelberg equilibrium (SE), the leader is better off and the follower is worse off than in a Cournot equilibrium (CE).
Stackelberg pointed out history, institution, law, discovery, and development as factors affect the determination of the leader and the followers in general industries. But in an electricity market, the transmission network is an important factor. Because the physical limits of transmission lines can restrict the economic dispatch of the generation power, the generation firms change their strategies by depending on the site with respect to the congested line. This paper postulates that the firm at the receiving area of a congested line has a leading opportunity, and the firm at the sending area has a follower position. When it is not clear if a firm locates in a sending area or a receiving area, PTDF(Power Transfer Distribution Factor) is a useful index to clear the vagueness
[14
,
18
,
19]
.
Let’s look into the duopoly system and derive the Stackelberg equilibrium (SE). Firms have generating marginal cost;
MC_{i}
=
m_{i}q_{i}
, and the inverse demand functions of the market is
p_{i}
=
a

r
(q
_{1}
+q
_{2}
). The firm F
_{1}
is assumed a leader, F
_{2}
follower. The follower’s best response condition is as follows;
The leader’s best response condition is derived by considering the follower reaction.
where, ∂q
_{2}
⁄ ∂q
_{1}
is obtained from (8).
The SE is derived from (7), (8) and the demand function is as follows;
where, Δ
_{A}
= m
_{1}
m
_{2}
+ 2r(m
_{1}
+ m
_{2}
) + 2r
^{2}
, Δ
_{B}
= m
_{1}
m
_{2}
+ r(2m
_{1}
+ m
_{2}
) + r
^{2}
. q
_{s1}
is the leader’s output, q
_{s2}
is the follower’s output, and p
_{s}
is the market price. The subscript‘s’ denotes for the SE and the subscript ‘c’ is used for the NCE.
where, Δ
_{A}
and Δ
_{B}
are the same as before.
 3.2 Social welfare comparison between NCE and SE
The NCE and the SE are compared from the viewpoint of social welfare which is used universally as an index for the trading value in a market in microeconomics. This section shows that social welfare in the SE is greater than in the NCE in no congestion situation by comparing price and total quantity supplied.
The ratio of the prices between the NCE and the SE is as follows;
where, Δ
_{A}
> Δ
_{B}
=Δ
_{A}
– r · m
_{2}
– r
^{2}
. Therefore, the price in SE is lower than that in the NCE.
The generation quantity of F
_{1}
in the SE and the NCE is compared by the following ratio;
The total quantity of generation, q
_{1}
+q
_{2}
, is derived from (8) in the SE and the NCE in common as follows;
The total quantity is proportional to the leader’s output, q
_{1}
. Since the quantities, q
_{s1}
>q
_{c1}
from the previous equation, the total quantity in the SE is greater than in the NCE. From the viewpoint of “Benefit” in demand function in microeconomics, the SE is better than the NCE, because the price in the SE is lower and the total demand in the SE surpasses that in the NCE. But the generation cost in the SE is larger than in the NCE because of greater demand in the SE.
Instead of the benefit in demand, the social welfare is suitable to evaluate a trade in a market. Since the social welfare is computed by benefit minus generation cost, the greater quantity may result in less social welfare.
Let’s compare quantity A and quantity B in
Fig. 4
. Even though the higher quantity B gives larger benefit, the social welfare in B is less than in A. The social welfare increases with increasing quantity until point C. On the other hand it decreases with increasing quantity after point C. So it is the point at issue whether the SE with greater quantity locates in region less or greater than C.
Relation between market price and marginal cost
Diagram of 3bus system with a congested line
In the region A less than C, the market price(p
_{a}
) is higher than the marginal cost(
MC
_{a}
). On the contrary, p
_{b}
is lower than
MC
_{b}
in region B. Therefore the prices in the SE need to compare with the marginal cost of the two firms for observing the SW.
For an easy comparison, the market price(p
_{s}
) derived in 3.1 is rearranged here;
Both the leader and the follower have lower marginal cost than the market price in the SE. So it is shown that the SE with a lower price and a larger total quantity has greater social welfare than in the NCE.
4. Results and Comparison
 4.1 Equilibrium in duopoly competition
 A. Uncongestion Case
In the duopoly system as
Fig. 1
, letting the transmission capacity be limitless leads to the NCE and the SE as in
Table 2
. In the SE, more demand quantity is supplied at a lower price than in the NCE. Therefore the SE gives more social welfare 7271.6 rather than 7188.6 of the CNE.
Results of NCE and SE in duopoly system without congestion
Results of NCE and SE in duopoly system without congestion
 B. Congestion Case
When the limit value, T, affect the strategies of the generation companies, the equilibria of Cournot and Stackelberg in
Table 2
will not be retained. The physical limit of the power flow on a transmission line might be advantageous to a potential supplier. The change of the NCE and the SE with a consideration of T=5.0 is in the
Table 3
.
Results of NCE and SE in duopoly system considering congestion
Results of NCE and SE in duopoly system considering congestion
As mentioned in 2.2 in this paper, taking a consideration of the limit T=5.0 results in “cycling” of firms’ choices instead of a pure equilibrium. To avoid the cycling phenomenon, a mixed strategy of the NCE is to be solved, and the mixed strategy equilibrium of the duopolyis given with probability α=0.657, β=0.343 for uncongestion and congestion case respectively. In the uncongestion case, the power flow is 1.53 less than T, and the price is identical at both buses. In the congestion case, the flow is equal to T, and the price at bus 1 is 55.42, the price at bus 2 is 50.11. The social welfare is calculated as an expectation value with the probability.
In the SE, the state corresponds to the case 2 in
Table 1
about the cycling reactions. From the viewpoint of the NCE, the firm F
_{1}
assumes F
_{2}
keeps the quantity q
_{2}
=49.58, and changes q
_{1}
from 39.58 to 53.72 as in
Table 1
. However, in Stackelberg, F
_{1}
believes that F
_{2}
keeps following F
_{1}
’s choice on the basis of F
_{2}
’s profit maximization. No cycling happens in the SE as in
Table 3
, and the power flow is equal to the limit T with identical prices. So this state is a borderbetween uncongestion and congestion. In the case of
Table 2
, the profit of F
_{1}
is computed to 1730 in the SE, however, in the case of
Table 3
, it is 1880.2. The leader, F
_{1}
, can increase its profit by utilizing the equipment scarcity of transmission capacity.
In the case of
Table 3
, the social welfare in the SE is 6369.5 lower than 6851.6 in the NCE. This paper shows that social welfare is higher in the SE than in the NCE as in
Table 2
, but when transmission congestion is considered it may not be true as in
Table 3
. The next section shows another result that the SE gives higher social welfare than the NCE even under consideration of congestion. The verification of the equilibria is provided in appendix A.
 4.2 Equilibrium in 3Players’ competition
An electricity power market is given with a simple network as shown in
Fig. 4
. The system consists of 3 generation firms, and 3 buses with a local market at each bus. The marginal cost functions of firm F
_{1}
, F
_{2}
, F
_{3}
, and the inverse demand functions are respectively,
MC
_{1}
= 10 + 0.3
q
_{1}
,
MC
_{2}
= 20 + 0.4
q
_{2}
,
MC
_{3}
= 15 + 0.45
q
_{3}
,
p
_{1}
= 70 − 0.7
d
_{1}
,
p
_{2}
= 80 − 0.5
d
_{2}
,
p
_{3}
= 90 − 0.4
d
_{3}
. The transmission lines are assumed to be lossless and have the reactance satisfying
x
_{12}
=
x
_{13}
= 2
x
_{23}
.
The transmission limit is assumed T
_{max}
=15.0, and the flow direction from 1 to 2 is determined by the inverse demand function at each bus and the marginal cost functions. The results of a mixed strategy of the NCE and a pure strategy of the SE are given in
Table 4
and verified in appendix B.
Results of NCE and SE in 3 firm competition systems under congestion
Results of NCE and SE in 3 firm competition systems under congestion
Due to the transmission limit, the NCE does not have a pure strategy, rather a mixed strategy. The uncongestion strategy in the MSNE has a probability 0.49. On the other hand, the congestion strategy has a probability 0.51, and leads to the nodal prices 52.3, 54.3 and 53.7 in bus 1, 2, and 3 respectively. The node of the highest price is bus 2, and it is a receiving end on the congestion line. Therefore the F
_{2}
that locates in bus 2 plays a role of a leader and the F
_{3}
and F
_{1}
are the followers.
This market system also gives a pure strategy of the SE. The SE state corresponds to a border between congestion and uncongestion states. The identical price at each node denotes an uncongestion state, and the power flow equal to the limit means a congestion state. The total generation is 173.93 in the SE, which is higher than 172.3 in the NCE computed using an expected value with the probabilities. The price in SE is 52.47, which is lower than 52.8 in the NCE calculated using weighted average over the buses and utilizing expected value with the probabilities. The more quantity and lower price leads to a higher social welfare in the SE than in the NCE.
In this 3player market, whether congestion arises or not,the social welfare in the SE is 7374.6 higher 7329.2 than in the NCE. This is one of the reasons that this paper recommends Stackelberg model for analyzing an electricity market with congestion. As mentioned before, the main reason is that the SE does not have a mixed strategy, while the NCE may have a mixed strategy. In order to implement this idea to the electricity market, amarked effort is needed to implement the concept of leader follower in the electricity market and to adjust the market rule to acknowledge the firstmover advantage for the firms in the receiving area of the congested line.
5. Conclusion
Transmission congestion may cause a mixed strategy equilibrium based on the Cournot model in an electricity market. A mixed strategy equilibrium is complicated to understand, difficult to compute, and hard to implement in a practical market. In order to avoid the mixed strategy Nash equilibrium, the Stackelberg model with a concept of leaderfollower is suggested and analyzed in this paper. The leader’s advantage corresponds to a major influence to congestion of a generation firm which is at the receiving end. The Stackelberg equilibrium is identified not to have mixed strategies, and have superiority to the Cournot Nash equilibrium from the viewpoint of social welfare in an uncongestion situation. When there is congestion in a transmission network, the Stackelberg equilibrium may have less social welfare. However, congestion is less frequent than uncongestion, so the SE will give better social welfare. This analysis will help to suppress bothersome arguments about mixed strategy equilibria in an electricity market.
BIO
KwangHo Lee He received his B. S., M. S., and Ph. D. degrees from Seoul National University in 1988, 1990, and 1995, respectively, in Electrical Engineering. He conducted research on reliability enhancement of power systems in the Korea Electrical Power Research Institute. He is presently a Professor in the Department of Electrical and Electronics Engineering at Dankook University, Yongin City, Korea.
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